Real Analysis 1 - Spring 2006


Henri Lebesgue, 1875-1941

COURSE: MATH 5210

TIME: 11:15-12:35 TR, PLACE: Room 477 of Brown Hall

INSTRUCTOR: Dr. Robert Gardner, OFFICE: Room 308F of Gilbreath Hall

OFFICE HOURS: TBA PHONE: 439-6979 (Math Office 439-4349)

E-MAIL: gardnerr@etsu.edu
WEBPAGE: http://www.etsu.edu/math/gardner/gardner.htm

TEXT: The primary text is Real Analysis with an Introduction to Wavelets and Applications by Don Hong (ETSU), Jianzhong Wang (Sam Houston State University), and Bob Gardner (ETSU), Elsevier Press, 2005.

ABOUT THE COURSE: This class offers a standard introduction to the theory of functions of a real variable from the measure theoretic perspective. Whereas the undergraduate real analysis class presents the results of calculus from a rigorous perspective, we will introduce fundamentally new ideas which are basic extensions of the results from calculus. In particular, we will put a weight or "measure" on certain sets of real numbers. This measure will be used to define a new type of integral called the Lebesgue integral. Recall that a function is Riemann integrable if and only if it is discontinuous on a "small" set (namely, a set of measure zero). The Lebesgue integral is much more flexible and will allow us to integrate a much larger class of functions.

GRADING: We will have two tests (T1 and T2) and homework (HW) will be taken up a regular intervals (weekly). Your average will be computed as follows:

AVERAGE = (T1 + T2 + 2HW)/4.
Grades will be assigned based on a 10 point scale with "plus" and "minus" grades being assigned as appropriate.

TENTATIVE OUTLINE:

Chapter 1: Fundamentals
Chapter 2: The Theory of Measure
1.1 Elementary Set Theory
2.1 Classes of Sets
1.2 Relations and Orderings
2.2 Measures on Rings
1.3 Cardinality and Countability
2.3 Outer Measure and Lebesgue Measure
1.4 The Topology of Rn
2.4 Measurable Functions
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2.5 Convergence of Measurable Functions
Chapter 3: The Lebesgue Integral
Chapter 4: Special Topics of Integration
3.1 Riemann Integral and Lebesgue Integral
4.1 Differentiation and Integration
3.2 The General Lebesgue Integral
4.2 Mathematical Models for Probability
3.3 Convergence and Approximation
4.3 Convergence and Limit Theorems
3.4 Lebesgue Integrals in the Plane
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Homework
Section
Problems
Due Date
Points
1.1
3, 5b, 6a, 6b, 10 (choose 3)
Tuesday, January 24
3+3+3=9
1.2
5b, 16, 18, 21 (choose 3)
Tuesday, January 31
3+3+3=9
1.3
1, 9, 10, 13 (choose 2); BONUS: 11, 14, 15
Tuesday, February 7
3+3+(3+4+4)=6+(11)
1.4
2, Problem 1, 7
Tuesday, February 14
3+3+3=9
1.4
16, 20, 24, BONUS: Problem 2
Tuesday, February 21
3+3+3+(3)=9+(3)
2.1
1c, 2a, 2b
Tuesday, February 28
3+3+3=9
2.2
2, 6, 7
Tuesday, March 14
3+3+3=9
2.3
5, 7, 9
Thursday, March 30
3+3+3=9
2.4
2a, 2b, 2c, 2d; BONUS: 7
Thursday, April 6
3+3+3+3+(3)=12+(3)
2.5
3, 5
Thursday, April 13
3+3=6
3.1
2, 3(ii), 3(iii), 4
Thursday, April 20
3+3+3+3=12
3.2
1, 2, 9
Thursday, April 27
2+3+3=8
3.3, 3.4
3.3 = 2, 9, 11; 3.4 = 1, 4, 5 (pick 4)
Tuesday, May 2
3+3+3+3=12
TOTAL
-
-
119+(17)
The numbers in parentheses represent bonus problems.

Problems

SOME SUPPLEMENTAL REFERENCES

"The Continuum Hypothesis, Part I" by W. Hugh Woodin, in Notices of the AMS volume 48, number 6, June/July 2001, 567-576. (The second part of the article appears in the following issue of the Notices). This article gives a bit of history of the Continuum Hypothesis and discussion of its relationship to various systems of axioms of set theory.

Everything and More: A Compact History of Infinity (Great Discoveries) by David F. Wallace. W. W. Norton & Company, October 2003. Although the primary aim of the book is to address the work of Cantor, there is also a lot of information on the history of analysis (and even a proof that a countable set has measure 0).

Introduction to Set Theory, K. Hrbacek and T. Jech. Marcel Dekker, 1999. A nice readable introduction to axiomatic set theory.

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